On 𝑝-torsions of geometric Brauer groups

doi:10.1515/crelle-2026-0087

DOI: 10.1515/crelle-2026-0087 ISSN: 0075-4102

On 𝑝-torsions of geometric Brauer groups

Zhenghui Li, Yanshuai Qin, Veronika Ertl

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Abstract

Let 𝑋 be a smooth projective integral variety over a finitely generated field 𝑘 of characteristic

p > 0

p>0

. We show that the finiteness of the exponent of the 𝑝-primary part of

Br ⁢ ( X k s ) G k

\mathrm{Br}(X_{{k^{s}}})^{{G_{k}}}

is equivalent to the Tate conjecture for divisors, generalizing D’Addezio’s theorem for abelian varieties to arbitrary smooth projective varieties. In combination with the Leray spectral sequence for rigid cohomology derived from the Berthelot conjecture recently proved by Ertl–Vezzani, we show that the cokernel of

Br nr ⁢ ( K ⁢ ( X ) ) → Br ⁢ ( X k s ) G k

\mathrm{Br}_{\mathrm{nr}}(K(X))\rightarrow\mathrm{Br}(X_{{k^{s}}})^{{G_{k}}}

is of finite exponent. This completes the 𝑝-primary part of the generalization of Artin–Grothendieck’s theorem on relations between Brauer groups and Tate–Shafarevich groups to higher relative dimensions.

Outline

On 𝑝-torsions of geometric Brauer groups

Abstract

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